02 analog filters - الصفحات الشخصية | الجامعة الإسلامية...
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Chebyshev Filters
Spring 2008© Ammar Abu-Hudrouss -Islamic
University Gaza
Slide ٢Digital Signal Processing
CN(x) is N th order Chebyshev polynomial defined as
Chebychev I filter
22
2
11)(NC
H
It has ripples in the passband and no ripple in the stop band
It has the following transfer function
1)coshcosh(
1)coscos(1
1
xxN
xxNxCN
The Chebyshev polynomial can be generated by recursive technique
11 2 kkk CCC
10 and1 CC
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Slide ٣Digital Signal Processing
Chebyshev polynomial characteristics
Chebyshev polynomial
1allfor1 NC
NCN allfor11
11intervaltheinoccursofrootstheAll xCN
even
11
odd10
2
2
N
NH
22
111
H
The filter parameter is related to the ripple as shown in the figures
Slide ٤Digital Signal Processing
Filter roots
The poles of LP prototype filter H (p ) are given by
Nkjp kkk ,,3,2,1coscoshsinsinh 22
1sinh1 12 N
N
kk 2
12
And the order of the filter is given by
s
AA ps
N
1
10/10/1
cosh110/110cosh
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Slide ٥Digital Signal Processing
p = 2500 rad/s , s = 12.5 Krad/s, For the LP prototype filter As = 30 dB, Ap = 0.5 dB
p = p / p = 1 rad/ss = s / p = 5 rad/s
Chebyshev filter type I
Example: Design lowpass Chebyshev filter with maximum gain of 5 dB. Ap = 0.5-dB ripple in the passband, a bandwidth of 2500 rad/sec. A stopband frequency of 12.5 Krad/sec, and As = 30 dB or more for > s
32676.2
)5(cosh110/110cosh
1
05.031
N
To determine the order of the filter
34931.05.0)1log(10 2
To calculate the ripple factor
Slide ٦Digital Signal Processing
Chebyshev filter type I
The transfer function626456.0and02192.1313228.0 jpk
To determine H0 we know that H (p = 0) = 1 (N is odd)
591378.034931.01sinh
21 1
2
62645.0142447.1626456.020
pppHpH
62645.0142447.11 0H 715693.00 H
And to have 5 dB gain we multiply H0 by 1.7783
62645.0142447.1626456.0
7783.1715693.02
ppp
pH
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Slide ٧Digital Signal Processing
Chebyshev filter type I
When we substitute by p =s/2500
156610714156610886.19
62
12
sss
sH
Slide ٨Digital Signal Processing
Chebyshev filter type II
This class of filters is called inverse Chebyshev filters. Its transfer function is derived by applying the following
1) Frequency transformation of ’ = 1/ in H (j ) of lowpass normalized prototype filter to achieve the following highpass filter
2) When it subtracted from one we get transfer function of Chebyshev II filter
'/111
'1
22
2
nCj
H
'/111
1'/11
1122
22
n
n
CC
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Slide ٩Digital Signal Processing
Chebyshev filter type II
It has ripples in the passband and ripple in the stop band
It has the following transfer function
Slide ١٠Digital Signal Processing
1. The frequencies are normalised by s instead of p.
2. The ripple factor i is calculated by
2 . The order of the filter is given by
3. The poles of LP prototype filter H (p) are given by
Chebyshev filter type II Design
Nkjp kkk ,,3,2,1coscoshsinsinh/1 22
1sinh1 12 N
N
kk 2
12
p
AA ps
N
/1cosh110/110cosh
1
10/10/1
110/1 1.0 sAi
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Slide ١١Digital Signal Processing
5. The zeros of H ( p ) is given by
The filter transfer function is calculated by
6. Restore the magnitude scale by calculating H0.
7. Restore the frequency scale to by frequency transformation
Chebyshev filter type II Design
n
k k
m
k k
pp
pHpH
1
120
20
2/,,3,2,1sec0 Nmkjz kkk
Slide ١٢Digital Signal Processing
p = 1000 rad/s , s = 2000 krad/s, For the LP prototype filter
p = p / s = 0.5 rad/ss = s / s = 1 rad/s
Chebyshev filter type II
Example: Design lowpass Chebyshev II filter that has 0-dB ripple in the passband, p = 1000 rad/sec, Ap = 0.5dB, A stopband frequency of 2000 rad/sec and As = 40 dB.
582.4
)2(cosh110/110cosh
1
05.041
N
To determine the order of the filter995.99/1
40)1log(10 2
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Slide ١٣Digital Signal Processing
Chebyshev filter type II
The transfer function
660.78777026-and
,11j0.485389050.52479948-,750.6108703160.15595592 jpk
.1.05965847995.99sinh51 1
2
60.78770266470.511016981.04959897
760.39747221520.311831181.70131.0515
2
2
22220
ppppp
ppHpH
The zeros is calculated by
2/,,2,1sec0 nmkjjz kkk
1.70131.0515 21 jzjz
Slide ١٤Digital Signal Processing
Chebyshev filter type II
H (s) can be achieved by substituting by p =s/2000
To determine H0 we know that H (p = 0) = 1 (N is odd)
15999426.02002.31 0H 0.0499950 H
0.1599942665725515054997130818905214913282
2002.304.42345
240
p.p.p.p.p
ppHpH
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Slide ١٥Digital Signal Processing
Elliptic filter is more efficient than Butterworth and Chebychev filters as it has the smallest order for a given set of specifications
The drawback it suffers from high nonlinearity in the phase, especially near the band edges.
Elliptic filter
The filter order requires to achieve a given set of specifications in passband ripple 1 and 2